Seems as though the socket may be faulty, loose or have a bad connection to the motherboard.
What type of connection are you using?
If you are using a mini Toslink cord then try using a mini jack stereo audio plug and check if that works normally (or vise versa).
If it is the mini Toslink that you're having the problem with then it may be a socket connection problem with the motheboard.
If it is the mini audio jack that you're having the problem with then it may be the socket itself or a socket connection problem with the motheboard.
Here is a link to an ifixit How to disassemble a Bowers & Wilkins Zeppelin repair guide.
It is in German and essentially it describes how to open the device to repair a fuse so hopefully it may be of some help. (I couldn't locate any other teardown info or service manual for the device, you may have more luck)
Does the socket seem "loose" when you try to gently move it from side to side?
If it is soldered correctly there should be no movement.
Looking at the pictures you posted I'm wondering if there is a dry solder joint on one of the connections of the socket.
I can't magnify it enough to see if it is just a spare terminal or whether it should be soldered to a track on the systemboard. It would explain the fact that applying pressure to the right side of the socket would make it work. There may be just enough movement for the connection to be made.
I've highlighted the suspect connection in the image below.
(click on image to enlarge for better viewing)
I may be totally wrong about this of course.